simpily `((3125)/(243))^(4//5)`

Answer

To simplify the expression \(\left(\frac{3125}{243}\right)^{\frac{4}{5}}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \left(\frac{3125}{243}\right)^{\frac{4}{5}} \]
We can rewrite the exponent \(\frac{4}{5}\) as:
\[ \frac{4}{5} = 4 \times \frac{1}{5} \]
Thus, we can express the original expression as:
\[ \left(\frac{3125}{243}\right)^{4 \times \frac{1}{5}} = \left(\left(\frac{3125}{243}\right)^{\frac{1}{5}}\right)^{4} \]

### Step 2: Calculate the fifth root
Next, we need to calculate \(\left(\frac{3125}{243}\right)^{\frac{1}{5}}\). We can factor \(3125\) and \(243\):
- \(3125 = 5^5\)
- \(243 = 3^5\)

So, we can rewrite the fraction:
\[ \frac{3125}{243} = \frac{5^5}{3^5} \]

### Step 3: Apply the fifth root
Now we apply the fifth root:
\[ \left(\frac{5^5}{3^5}\right)^{\frac{1}{5}} = \frac{5^{5 \cdot \frac{1}{5}}}{3^{5 \cdot \frac{1}{5}}} = \frac{5^1}{3^1} = \frac{5}{3} \]

### Step 4: Raise to the power of 4
Now we raise \(\frac{5}{3}\) to the power of \(4\):
\[ \left(\frac{5}{3}\right)^{4} = \frac{5^4}{3^4} \]

### Step 5: Calculate \(5^4\) and \(3^4\)
Calculating \(5^4\) and \(3^4\):
- \(5^4 = 625\)
- \(3^4 = 81\)

Thus, we have:
\[ \frac{5^4}{3^4} = \frac{625}{81} \]

### Final Answer
Therefore, the simplified form of \(\left(\frac{3125}{243}\right)^{\frac{4}{5}}\) is:
\[ \frac{625}{81} \]

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